6 men complete a piece of work in 12 days. 8 women can complete the same piece of work in 18 days. Whereas 18 children can complete the piece of work in 10 days. 4 men, 12 women and 20 children work together for 2 days, and then only 36 men were to complete the remaining work in x day.
A can do a piece of work in 10x days and B can do the same work in 20x days. They do the work on alternative days starting from A then in how many days A and B can complete the work?

Options

A. 11 days
B. 12 days
C. 13 days
D. 14 days

Answer

To solve the given problem step by step, we will first find the efficiencies of men, women, and children, calculate the total work done, and then determine how many days A and B will take to complete the work together.

### Step 1: Calculate the efficiency of men, women, and children

1. **Men's Work**:
- 6 men complete the work in 12 days.
- Total work = 6 men × 12 days = 72 man-days.
- Therefore, 1 man’s work in 1 day = 1/72 of the work.

2. **Women's Work**:
- 8 women complete the work in 18 days.
- Total work = 8 women × 18 days = 144 woman-days.
- Therefore, 1 woman’s work in 1 day = 1/144 of the work.

3. **Children's Work**:
- 18 children complete the work in 10 days.
- Total work = 18 children × 10 days = 180 child-days.
- Therefore, 1 child's work in 1 day = 1/180 of the work.

### Step 2: Find a common efficiency ratio

To find a common efficiency ratio, we can set the work done by men, women, and children equal to a common value:

Let:
- Efficiency of 1 man = M
- Efficiency of 1 woman = W
- Efficiency of 1 child = C

From the calculations:
- 6M × 12 = 720 (total work)
- 8W × 18 = 720 (total work)
- 18C × 10 = 720 (total work)

From these equations, we can derive:
- M = 10
- W = 5
- C = 4

### Step 3: Calculate the total work done by 4 men, 12 women, and 20 children in 2 days

1. **Work done by 4 men in 1 day**:
- 4 men = 4 × 10 = 40 work units.

2. **Work done by 12 women in 1 day**:
- 12 women = 12 × 5 = 60 work units.

3. **Work done by 20 children in 1 day**:
- 20 children = 20 × 4 = 80 work units.

4. **Total work done in 1 day**:
- Total = 40 + 60 + 80 = 180 work units.

5. **Total work done in 2 days**:
- Total work in 2 days = 180 × 2 = 360 work units.

### Step 4: Calculate the remaining work

1. **Total work**: 720 work units.
2. **Work completed in 2 days**: 360 work units.
3. **Remaining work**: 720 - 360 = 360 work units.

### Step 5: Calculate how many days 36 men will take to complete the remaining work

1. **Efficiency of 36 men**:
- 36 men = 36 × 10 = 360 work units in 1 day.

2. **Days required to complete remaining work**:
- Remaining work = 360 work units.
- Days = 360 / 360 = 1 day.

### Step 6: Determine the value of x

Since the remaining work can be completed in 1 day by 36 men, we have:
- x = 1.

### Step 7: Calculate how many days A and B can complete the work

1. **A's efficiency**: A can do the work in 10x days = 10 days.
- Work done by A in 1 day = 1/10 of the work.

2. **B's efficiency**: B can do the work in 20x days = 20 days.
- Work done by B in 1 day = 1/20 of the work.

3. **Combined work in 2 days**:
- Day 1 (A): 1/10
- Day 2 (B): 1/20
- Total work in 2 days = 1/10 + 1/20 = 2/20 + 1/20 = 3/20 of the work.

4. **Total work to be done**: 1 (whole work).
5. **Number of 2-day cycles needed**:
- Total cycles = 1 / (3/20) = 20/3 cycles = 6.67 cycles.
- This means they will take 13 days to complete the work since A will start the next day after 12 days.

### Final Answer
A and B can complete the work in **13 days**.